Discard mesh zones from solidify modifier
You can choose which part you want to solidify by setting Vertex Groups.


Go to Edit Mode (TAB) and select vertices you want to solidify.
Go to Data tab and under Vertex Group section add ('+' sign) new group and choose Assign (with vertices still selected).





Go back to Object Mode (TAB) and add Solidify Modifier.
Under it's option choose created vertex group.





Set Thickness and a little bit of Factor not to have Z fighting problem.